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Solid geometry: prisms, pyramids, spheres

Volumes, surface areas and cross-sections in three dimensions.

Prisms and cylinders are base × height; pyramids and cones a third of that; spheres 4/3 πr³. Picture it: a cone fits three times into the cylinder around it. Rotate the 3D view to see the base and slant. Think it: Cavalieri's principle (equal cross-sections at every height means equal volume) is the idea calculus later makes exact.

Exemple résolu · volume of a cone with radius 3 height 4

Geometry · Solids

Volume of a cone with radius 3 height 4

Étape par étape

  1. r = 3,\ h = 4,\ \ell = \sqrt{r^2+h^2} = 5

    Cone\; ℓ is the slant height.

  2. V = \tfrac{1}{3}\pi r^2 h = 12 \pi \approx 37.699

    Volume: a third of the cylinder around it.

  3. S = \pi r^2 + \pi r \ell = 24 \pi \approx 75.398

    Surface area = base + lateral.

Révèle la réponse
V = 12 \pi \approx 37.699,\quad S = 24 \pi

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Comment faire: Solid geometry: prisms, pyramids, spheres

  1. Cone\; ℓ is the slant height.
  2. Volume: a third of the cylinder around it.
  3. Surface area = base + lateral.

Questions posées par les gens

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

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